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Quantum Combinatorial Games Paul Dorbec1,2 Mehdi Mhalla3 1 Univ. Bordeaux, LaBRI, UMR 5800, F-33400 Talence, France 7 2 CNRS, LaBRI, UMR 5800, F-33400 Talence, France 1 3CNRS, LIG, Université Grenoble Alpes, France 0 2 January 10, 2017 n a J 9 Abstract In this paper, we propose a Quantum variation of combinatorial games, generalizing the ] M Quantum Tic-Tac-Toe proposed by Allan Goff [2006]. A Combinatorial Game is a two-player gamewithnochanceandnohiddeninformation,suchasGoorChess. Inthispaper,weconsider D thepossibilityofplayingsuperpositionsofmovesinsuchgames. Weproposedifferentrulesets . depending on when superposed moves should be played, and prove that all these rulesets may s lead similar games to different outcomes. We then consider Quantum variations of the game c [ of Nim. We conclude with some discussion on the relative interest of the different rulesets. 1 v 1 Introduction 3 9 Quantuminformationtheoryanditsinterpretationsrequiredtheintroductionofnewconceptssuch 1 2 as superposition, entanglement, etc. To shed a new light on these concepts, it seems relevant to 0 presentthemwithintheframeofagame. Moreover,introducingthesephenomenaincombinatorial . game theory induces new families of games that had no natural motivation beforehand. 1 0 AQuantumvariationofTic-Tac-Toewasproposedin[2],“asametaphorforthecounterintuitive 7 nature of superposition exhibited by quantum systems”. In this paper, we want to propose a 1 general way to provide a Quantum variation of any game. Moreover, we want to improve the : v game interpretation of the measurement, in order to make it closer to what happens in quantum i informationtheory. Themainideaconsistsinallowingaplayertoplay“asuperposition”ofmoves. X Nevertheless, there are various ways to introduce Quantum variations of a combinatorial game in r a general, we consider some of them and argue on the interest of the different interpretations. After defining some useful notions from classical combinatorial game theory, we propose a definition of Quantum variations of a game, with different rulesets. We then prove in Section 4 that each pair of rulesets differ for some game. Finally, we study the different rulesets on the game of Nim which is a classical game for combinatorial game theory. 1 2 Reminders on Classical Combinatorial Game Theory A combinatorial game is a game with no chance and no hidden information, opposing two players, Left and Right. In Combinatorial Game Theory (CGT), a game is described by a position (e.g. the setting of the pieces on a board) and a ruleset, that describes the legal moves for each position and the positions to which these moves lead. We here consider short games, that is games with no possible infinite run and a finite number of moves. A classical example of such games is the game of Nim, that we consider often in the following. It is a two player game played on heaps of tokens. Each player alternately take any (positive) number of tokens from a single heap. When a player is unable to move (i.e. no heaps have any token left), he loses. To have a synthetic description of a game, a move is generally described in a general setting, independent of the current position (e.g. a bishop moves along a diagonal in Chess). In the following, we need a description of a move that can be interpreted with no ambiguity in different positions of the game. In a formal way, we describe a combinatorial game with a set G of positions and an alphabet Σ of all possible moves. Then a Ruleset is defined as a mapping : Γ:GˆΣÑGYtnullu where null means that the move is not legal from that position in the game. For example, in the game of Nim, we denote Nimpx ,...,x q the position with k heaps of 1 k x ,...,x tokens. If the heaps contain at most n tokens, the alphabet of moves can be defined by 1 k pairs of integers: Σ“t1,...,kuˆtń,...,´1u, a pair pi,´jq meaning we try to remove j tokens from heap number i. (In Partisan games where players have different moves, the moves mPΣ are differentiated by their player.) Under normal convention (that we follow here), any player that is unable to move from a positionloses. Acombinatorialgame(opposingtwoplayers,Left andRight)mayhavefourpossible outcomes: N when the first player can force a win, P when the second player can force a win, L when Left can force a win no matter who moves first and R when Right can force a win no matter who moves first. See [5] for more details. Another fundamental concept in classical CGT is the sum of two games. When playing on a sum, each player plays a move in one of the summand, until no moves is legal in any game. With our previous setting, from two rulesets Γ :G ˆΣ ÑG Ytnullu 1 1 1 1 Γ :G ˆΣ ÑG Ytnullu 2 2 2 2 we define the ruleset Γ of the sum (assuming Σ XΣ “H) by 1`2 1 2 Γ : G ˆG ˆpΣ YΣ qÑG ˆG Ytnullu 1`2 1 2 1 $ 2 1 2 ’&pΓ pG ,mq,G q if mPΣ and Γ pG ,mq‰null 1 1 2 1 1 1 pG1,G2,mqÑ’%pG1,Γ2pG2,mqq if mPΣ2 and Γ2pG2,mq‰null null otherwise. Recall that two games G and G are said to be equivalent, denoted G ” G , if for any third 1 2 1 2 gameG ,G `G asthesameoutcomeasG `G . ObservethatifalloptionsofagameΓ pG ,mq 3 1 3 2 3 1 1 are equivalent to the corresponding option of γ pG ,mq, the two games are equivalent. The value 2 2 2 of a game is its equivalence class. In particular, ˚k denotes the value of a single Nim heap of k tokens (we use notation with ˚0“0 and ˚1“˚). In impartial games under normal convention, all games are equivalent to a Nim heap. Then the value of a game can be computed from the values of its options by the mex-rule, i.e. the value of an impartial game is ˚x where x is the smallest non-negative integer such that ˚x is not among the options. The birthday of a game is the maximum number of moves that can be played on the game. 3 Definition of a Quantum game Figure 1: Two superposed moves played on a game of Nim. From a classical game, we define a Quantum variation of the game (that can be seen as a modified ruleset) where players have to play Q-moves. A Q-move consists in a superposition (seen as a set) of classical moves. If the Q-move is reduced to a single classical move, we call it an unsuperposed move. As soon as a superposition of moves is played, the game is said in Quantum state. For each sequence of Q-moves, a run is a choice of a classical move among each superposed move of the sequence. If the choice forms a legal sequence of moves in the corresponding classical game, then the position obtained after this run is called a realisation of the game. For example, in Figure 1, two superposed moves are played on a game of Nim, leading to only three realisations, since the sequence (1,-3)–(1,-2) is not legal. InaQuantumstate, aclassicalmoveisconsideredlegalifthereexistsatleastonerealisationof the game where this move is legal (in the classical sense). A superposition of moves is legal if each ofitssuperposedclassicalmovesarelegal(possiblyindifferentrealisations). Observethatapplying amovemakesdisappearallrealisationsinconsistentwiththatmove. Thegameendswhenthenext player has no legal moves. It should be noted here that with this setting, the way the Ruleset is defined influences the behaviour of its Quantum variation. This is why we cannot use simply the set of options of a position to describe a game, as it is in classical CGT. Moves need to be labeled and to have interpretations in different positions. In particular, if we choose to describe the possible moves by the resulting position, then the game would become much simpler, since the positions obtained by any legal run would depend only of the last move. 3 3.1 Quantum Rulesets definition Depending on when unsuperposed moves are considered legal, we here consider five rulesets. Ruleset A: Only superpositions of at least two (classical) moves are allowed. Ruleset B: Only superpositions of at least two moves are allowed, with a single exception: when the player has only one possible move within all realisations together, he can play it as an unsuperposed move. RulesetC:Unsuperposedmovesareallowedifandonlyiftheyarevalidinallpossiblerealisations. Ruleset C’: a player may play an unsuperposed moves if and only if it is valid in all possible realisations where he still has at least one classical move. Ruleset D: Unsuperposed moves are always allowed (seen as the superposition of two identical moves). Note that Ruleset C’ is the least permissive ruleset that is more permissive than Ruleset B and Ruleset C. We get the following lattice of permissivity: D C’ B C A Remark that in all the above rulesets, the availability of a move depends only on the existence of a realisation, independently on how the realisations are correlated. Therefore, the history may be ignored, implying the following observation. Observation 1 Independently of the sequence of moves, if two games are superpositions of the same realisations, the possible sequences of legal moves are identical. (In CGT, such games are said to be equal.) From this observation, we deduce that a position of a Quantum game is fully described by the possible realisations of the earlier moves, which form a multiset of classical positions. Since the multiplicityofaclassicalpositionisnotimportantinal7ltheabove?rulesets,wedescribesuchaposi- 7 ? tionasasetofsuperposedclassica7lposit?ions,denoted G1,...,Gn . Forexamplethesu7per?position 7 ? 7 ? of positions G and G1 is denoted G,G1 , and the unsuperposed position G as the set G . When w7 e discuss o?n a specific Ruleset, e.g. Ruleset A, we use the letter as a subscript to the notation: 7 ? G ,...,G . 1 n A 77 ?? Legal classical moves in a superposition G ,G ,...,G now are any mPΣ such that at least 1 2 n one of ΓpG ,mq is not null. In that case, we get the Quantized ruleset defined on PpGqˆPpΣq by i 7 ? 7 ? 7G ,...,G ?ˆtm ,...,m uÑ7ΓpG ,m q|1ďiďn,1ďj ďk,ΓpG ,m q‰null? 1 n 1 k i j i j 4 3.2 Sum of Quantum games and Quantum sum of games Note that given this setting, a Quantum combinatorial game can be seen as a combinatorial game withaparticularruleset. WecanthussumaQuantumcombinatorialgamewithanotherQuantum combinatorial game, or with a classical game, following the classical game sum (each player plays a move in one of the summand, until no moves is legal in any game). In such a sum of games, no superposition of moves distributed among the terms are allowed. However, we can also consider the Quantum variation of the sum of two games, thus allowing superposition of moves distributed among the operands of the sum. Then we get 7 ? 7 ? 7 ? 7 ? 7 ? 7 ? G ` H ‰ G`H Observe that the Nim product b, as defined in [1, 5], allows to play a move on one operand or on both. In Ruleset D, where allowed moves are superpositions of any number of classical moves, we then obtain 7 ? 7 ? 7 ? 7 ? 7 ? 7 ? G`H “ G b H . D 7 D ? D 7 ? This is not true for other Rulesets because in G`H ,7yo?u could play only one element of the 7 ? superposition in G while it might not be a legal move in G . 3.3 To play the game : If we want to play a Quantum variation of a game without computing all the realizations of the game at each move, it becomes difficult to know whether a move is legal or not. We suggest that each player at his turn announces the moves he wants to play. If his opponent suspects this move is not legal, he can challenge the player. Then the player must prove his move was legal: he must exhibit a run for each of the classical moves he played. If he manages to do so, then he wins, otherwise he loses. Note anyway that this does not change the outcome of the game: if a player can ensure a win in any of these settings, he can ensure a win in the other too. 3.4 Link with Quantum theory Quantum combinatorial games have the following interpretation in quantum theory : the players act on a quantum system. Each move of a player consists in applying a unitary operator to the system. If at some point, the move of some player brings the state of the system into a losing subspace, the player loses. To decide if the state is in the losing subspace, one may apply a projective measurement. This produces a probabilistic game where each player can at best ensure a probability of winning. To recoverthecombinatorialversion,weneedtoconsiderwinninginapossibilistic way: aplayerloses at the first step for which the projective measurement puts the state in the losing subspace with probability 1. This could be interpreted as allowing the player to replay the sequence of unitaries and the measure until he manages to prove his system is not in the losing subspace. Note that when a game is in a quantum state, the possible classical states in the superposition caninteractwitheachotherandasuperpositionofmovescanactondifferentelements. Therefore, likeinquantumtheory, beinginasuperpositionofpositionsisdifferentthanbeinginaprobability distribution over states. 5 4 Difference of rulesets We now give some examples showing that all the rulesets are non-equivalent for some games. Example 1 The first example is based on the octal game 0.6. This is a game played on a line of pins. Each player at his turn must remove a single pin that is not isolated (i.e. it still has at least one adjacent pin). We number the pins from 1 to n according to their position on the line. The octal game rule specifies that the move i becomes illegal when both ti´1,i`1u have been played. We consider game 0.6 played on a line of four pins. • the game has outcome P in the classical game. • it has outcome N in all other rulesets. A winning first move is p1,4q. Then whatever the answer to this move is, he can win playing 2, 3 or p2,3q. Example 2 Our second example is played on the game Domineering. This game is played on a subset of squares from a grid. Left plays vertical dominoes (two adjacent squares) and right plays horizontal dominoes. A move is described by the squares the played domino occupies. We play Domineering on the following board: • The game has outcome R in the classical game. Indeed, if Left plays first, Right can still play a horizontal domino, while Right can prevent Left from being able to play with his first move. Right can use the same strategy in Ruleset D. Right is the only one who can play a superposition of moves, that makes him win in Ruleset A. Moreover, playing his winning move in the classical game, he wins in Ruleset C and Ruleset C’. If Left starts in Ruleset C or Ruleset C’, he must play an unsuperposed move and Right can answer also with an unsuperposed move. • In Ruleset B Right playing first is forced to play the superposed move, to which Left can answer. Left still loses when playing first. Example 3 We consider the game called Hackenbush. We define here a restricted version that suffice to illustrate our example. The game is played on a set of vertical paths composed of blue andred1 edges. AthisturnLeftselectsablueedgeandremoveittogetherwithalltheedgesabove it. Right does the same with a red edge. The game ends when a player has no edge to select. We label each edge of the game, and define a move by the label of the edge selected. We play Hackenbush on the following game 2 3 1 a b 1Wedrawrededgeswithparallellines. 6 In that game, in Ruleset A, Right cannot play after Left played twice pa,bq and this is the only rulesetwhenLeftwinsplayingfirst. Thesequencep2,3q-paq-p1,3q-pbqallowsanextramovep1qonly in Ruleset C’ and Ruleset D. • the game has outcome P in the classical game as well as in Ruleset C. • it has outcome L in Ruleset A. • it has outcome R in Ruleset B, Ruleset C’ and Ruleset D. Example 4 We play Nim on two heaps of two tokens, i.e. Nimp2,2q. This is N in Ruleset D but P in Ruleset C’. The key observation is that the sequence pp1,´1q,p2,´1qq – pp1,´2q,p2,´1qq – p1,´2q is winning for the first player in Ruleset D but the last move is not allowed in Ruleset C’. 5 Results 7 ? 7 ? Definition 2 A game G in a superposition G ,G ,...,G is said to be covered by G ,...,G 1 1 2 n 2 n ifeitherG hasnolegal(classical)moves, orforeverymovemPΣthatislegalon G , thereexista 1 1 gameG ,2ďiďnsuchthatmislegalonG andΓ pG ,mqiscoveredbyΓ pG ,mq,...,Γ pG ,mq. i i 1 1 2 2 n n Observe for example that a Nim heap with k tokens is covered by a Nim heap with more than k tokens, which can easily be proved inductively. 7 ? 7 ? Theorem 3 In Ruleset A, Ruleset B and Rule7set D, if a super?pos7ition of gam?es G1,G2,...,Gn 7 ? 7 ? is such that G is covered by G ,...,G , then G ,G ,...,G ” G ,...,G . 1 2 n 1 2 n 2 n Proof: We prove the result by induction on the birthday of G . First observe that if G has 1 1 no possible moves, then the equivalence is clear in Ruleset A, Ruleset B and Ruleset D (but not in Ruleset C and Ruleset C’). Now suppose the result is true for every game with birthday less t7han G1’s birthday. Consider a superposed move pm?1,...,mkq. Then the game obtained is 7ΓpG ,m q,1 ď i ď n,1 ď j ď k,ΓpG ,m q ‰ null?. For each 1 ď j ď k such that i j i j ΓpG1,mjq‰null, ΓpG1,mjqiscoveredb7yΓ2pG2,mjq,.?..,ΓnpGn,mjq, soitcanberemovedfrom 7 ? the7superpositi?on. Thus,everyoptionof G1,G2,...,Gn isequivalenttothecorrespondingoption 7 ? of G ,...,G and we have equivalence. l 2 n From our above observation on the Nim heaps, we get: Corollary 4 In Rulesets A,B,D, the value of a superposition of a single Nim heap in different states depends only on the largest heap in the superposition. Corollary 5 InRulesetA,RulesetBandRulesetD,theNimgameonatmostk heapsisequivalent to the corresponding game where only superposition of at most k moves are allowed. Proof: We prove the result by induction on the birthday of the game: Consider a superposition of Nimheapsonatmostkheaps. Considerasuperposedmovefromthisgame,saypph,í q h ihPIh,1ďhďk (i.e. the superposed removal of all numbers of token in I for each heap h). Theorem 3 shows h thatthis move is equivalent to the move pph,´minpI qq q, which is a superposition of at most h 1ďhďk k moves. In the case when only one h is such that I is non empty (and unsuperposed moves are h not legal), we also get that it is equivalent to the move pph,´minpI q,ph,´minpI q´1qq. l h h 7 We remark that this corollary is not true in the case when the number of heaps is larger than k. For example, let A|2 denote7 Ruleset A?where only superposi7tions of two moves are a?llowed. 7 ? 7 ? We have that all the options of Nimp1,1,1q are equivalent to Nimp1,0,1q,Nimp0,1,1q (by A|2 77 ?? A|2 permutation of the heaps). This position has two options, namely Nimp0,0,1q (by the move 77 ?? A|2 p1,´1q,p2,´1qq) which has value 0 and Nimp1,0,0q,Nimp0,1,0q,Nimp0,0,1q (by any other 77 ?? 77 A|2?? move) which has value ˚. Thus Nimp0,1,1q,Nimp1,0,1q ”˚2 and Nimp1,1,1q ”0. A|2 77 A|2?? However p7laying the only possible superpositi?on of three moves on Nimp1,1,1q leads to 7 ? A the7position Nimp0,1,1q,Nimp1,0,1q,N?imp1,1,0q . All options of this position are equivalent 7 ? A 7to Nimp0,0,1q,Nimp0,1,0q,Nimp1,0,?0q which has va7lue ˚. As a?consequence, we obtain that 7 ? A 7 ? Nimp0,1,1q,Nimp1,0,1q,Nimp1,1,0q ”0 and finally Nimp1,1,1q ”˚. A A Intherestofthissection,wegetinterestedinthevaluesofQuantumsingleheapsofNimunder the different rulesets. 5.1 Values of single Nim heaps Lemma 6 In Ruleset A, the value of a superposition of Nim heaps has value ˚pk´1q if and only if its largest heap has size k ě0, i.e. 7 ? 7 ? Nimpi q,...,Nimpi q ”˚pk´1q where k “ maxpi q 1 (cid:96) A 1ďjď(cid:96) j 7 ? 7 ? Pr7oof: Weproveth?eresultbyinduction. Ifthelargestheapisofsize1,thenthegameis Nimp1q 7 ? or Nimp0q,Nimp1q . No superposed move are possible, thus the game has value 0. Assume now the lemma is verified up to k. Then we claim a position with as largest heap k`1 has value ˚k. First, all the legal moves are of the form p1,´x q for some 1 ď x ď k`1. Denote i i x “ minpx q. The move brings to a position whose largest heap is of size k´x`1, which have i value ˚pk´xq by induction. This proves the Lemma. l Lemma 7 In Ruleset B, the value of a superposition of Nim heaps of size k and less has value : $ 7 ? ’&0 if k “2 7 ? Nimpi1q,...,Nimpi(cid:96)q B ”’%˚ if k “1 where k “1mďjaďx(cid:96)pijq ˚pk´1q otherwise 7 ? 7 ? Pr7oof: Weproveth?eresultbyinduction. Ifthelargestheapisofsize1,thenthegameis Nimp1q 7 ? or 7Nimp0q,?Nimp1q . Thereisasinglelegalmove,whichistheunsuperposedmovep1,´1qthatlead 7 ? to Nimp0q . Thus the game as value ˚. If the largest hea7p is of size 2, th?en only the superposed 7 ? move pp1,´1q,p1,´2qq is legal, which brings to the game Nimp0q,Nimp1q of value ˚. Thus it has value 0. Assume now the lemma is verified up to k ě 2. Then we claim a position whose largest heap has size k `1 has value ˚k. First, all the legal moves are superpositions of the form p1,íq iPI with I Ă r1,k`1s. The move brings to a position whose largest heap is of size k´j `1 where j “mintiPIu, which has value ˚pkíq if kíě2, 0 if kí“1 and ˚ if kí“0, by induction. This proves the Lemma. l 8 Lemma 8 In Ruleset D, the value of a superposition of Nim heaps has value ˚k if and only if its largest heap has size k, i.e. 7 ? 7 ? Nimpi q,...,Nimpi q ”˚k where k “ maxpi q 1 (cid:96) D 1ďjď(cid:96) j Proof: Again the proof works by induction. In Ruleset D, a superposition with largest heap of size k has value at least ˚k since all unsuperposed moves are legal, and all its options have value at most ˚pk´1q by induction. l In contrast with what happens in the previous rulesets, the value of a superposition of Nim Heaps in Ruleset C does not depend only on the size of the maximum heap, as shows the following lemma. Lemma 9 In Ruleset C, a superposition of Nim heaps with maximum size k has value ˚pk´1q, unless there is only one element in the superposition in which case the game has value ˚k. i.e. # 7 ? 7 ? ˚k if (cid:96)“1 Nimpi q,...,Nimpi q ” where k “ maxpi q 1 (cid:96) C ˚pk´1q otherwise 1ďjď(cid:96) j Proof: Againtheproofisbyinductiononk “max pi q. IfNimp0qbelongtothesuperposition 1ďjď(cid:96) j (i.e. min pi q “ 0), no unsuperposed move is legal, then the options are all superposed. Also 1ďjď(cid:96) j ifthegameissuperposedwithsomethingelse,allmovesbringtoasuperposedgame. Inbothcases theoptionsaresuperpositionofatleasttwoNimheapsofanymaximumsizek1 with0ďk1 ďk´1, which have value ˚pk1´1q, so the game has value ˚pk´1q. Nowifthereisonlyoneelementinthesuperposition, allunsuperposedmovesarelegal, andthe values of the options span the interval r0,k´1s. l Lemma 10 In Ruleset C’, a superposition of Nim heaps with maximum size k has value ˚k unless Nimpk´1q be7longs to the sup?erposition. In that case, th7e game has valu?e ˚pk´1q, with the only 7 ? 7 ? exceptions of Nimp0q,Nimp1q which has value ˚ and Nimp1q,Nimp2q which has value 0. C1 C1 $ 7 ? 7 ? ’’’’’’&0oifrG77N“im7pN0iqm,Np1imq,pN1iqm,Np2imq?pC21q??C1 7 ? G“ Nimpi1q,...,Nimpi(cid:96)q C1 ”’’’’’’%˚˚pikf´k “1q1if k´1Ptiju where k “1mďjaďx(cid:96)pijq ˚k otherwise Proof: We first consider separatel7y the ga?mes w7ith a largest he?ap of size less than 3. Observe 7 ? 7 ? first that in Ruleset C’, the games Nimp1q and Nimp0q,Nimp1q have only7p1,´1q as a pos?sible 7 ? m7 ove, which leads to an?ended game. Thus they have value ˚. Now in both Nimp1q,Nimp2q and 7 ? 7Nimp0q,Nimp1q,?Nimp2q , the options are p1,´1q and pp1,´1q,p1,´2qq. Both lead to t7he game? 7 ? 7 ? Nim7p0q,Nimp1q wh?ich has value ˚7, so the?in7itial gam?e has7value 0. F?inally, the games Nimp2q 7 ? 7 ? 7 ? 7 ? and Nimp0q,Nimp2q have options Nimp0q , Nimp1q , and Nimp0,1q and thus have value ˚2. Considernowtheothergames, whichhavealargestheapinthesuperpositionofsizek ě3. We prove the lemma by induction on k. Suppose first that Nimpk´1q belongs to the superposition. Then the move pp1,´jq,p1,´j´1qq leads to a superposition of largest heap Nimpk´jq that also 9 containsNimpk´j´1q, thusofvalue˚pk´j´1qif1ďj ďk´3, 0ifj “k´2, and˚ifj “k´1. Moreover, all moves bring to a game with largest heap of size k1 with k1 ďk´1 and that contain Nimpk1´1q. AssumenowthatNimpk´1qdoesnotbelongtothesuperposition. Thenthemovepp1,´jq,p1,´kqq (for 1 ď j ď k ´2) leads to a superposition of largest heap Nimpk ´jq that does not contain N7 impk´j´1q,?so of value ˚pk´jq by induction. With pp1,´kq,p1,´k`1qq, we reach the game 7 ? Nimp0q,N7imp1q which has value?˚. With the move pp1,´kq,p1,´k `1q,p1,´k `2qq, we reach 7 ? the game Nimp0q,Nimp1q,Nimp2q which has value 0. All moves reach a game with value at most ˚pk´1q since the largest heap of the resulting game is of size at most k´1. This concludes the proof. l 6 Conclusion 6.1 Quantum variations of games with more than one heap. In the above, we restricted ourselves to the study of games with a single Nim heap, as an initial approach. The next step is naturally to consider games where there are two heaps or more. Such a game is fundamentally different from the sum of two games on one heap, since it becomes possible toplayasuperpositionofmovesondifferentheaps. Wethenhavetoconsidersomeextracaseslike in the following lemma. 7 ? 7 ? Lemma 11 In Ruleset A, for i,j ą 0, Nimpi,0q,Nimp0,jq ” ˚pmaxpi,jq´1`δ q where δ A i,j i,j denotes the Kronecker delta. 7 ? 7 ? Proof: Let S “ Nimpi,0q,Nimp0,jq , with i ě 0, j ą 0. Observe first that if i “ 0, j “ 1, then there is no possible move and the game has value 0. We now consider the possible moves on S. First observe that every possible move brings to a gamethatisequivalenttoasuperpositionofatmosttwogames. Indeed,7asuperpositionofclassical 7 move?s on the first heap pp1,i1q7,...p1,íkqq brings?to a superposition Nimpií1,0q,...Nimpi´ ? 7 ? i ,0q , which is equivalent to Nimpi´min ti u,0q by Theorem 3. This option has value ˚pi´ k l l minltilu´1q by7Lemma 6, which is at?most ˚pi´2q. Similarly, moves pp2,´j1q,...p2,´jkqq bring 7 ? to the position Nimp0,j ´minltjluq with value at most ˚pj ´72q. Finally, moves of the form 7 pp1,í1q,?...p1,ílq,p2,´j1q,...p2´jk1qleadtothesuperposition Nimpi´minltilu,0q,Nimp0,j´ ? min tj uq . l l By induction, these positions have value at most ˚pmaxti,ju´1q, and the only situation when thismaximumval7ueisattainediswhe?ni“j, andthemoveplayedispp1,´1q,p2,´1qq. Thisprove 7 ? that the value of Nimpi,0q,Nimp0,jq is at most ˚pmaxpi,jq´1`δ q. A i,j To show the other inequality, we simply need to observe that all the values ˚k for k ďj´2 are reached by the moves pp2,´j`kq,p2,´j`k`1qq, and similarly for i on the first heap. l 6.2 Discussion on the different rulesets. Among the proposed rulesets, we think that Ruleset C is interesting for its physical interpretation: when the position is classical we can use classical moves but when applying a superposed move we put the system in a superposition and we can no-longer act classically: one cannot force a branch of a superposition by choosing a classical move. 10